The lengths of the sides of a triangle are given in the ratio of 5:6:7. Let us denote the lengths of the sides as follows:
- 5x
- 6x
- 7x
Here, ‘x’ is a common multiplier that we need to find. The perimeter of the triangle can be calculated by adding all three sides:
Perimeter = 5x + 6x + 7x = 18x.
According to the problem, the perimeter is less than 54 cm. Thus, we can set up the following inequality:
18x < 54.
To solve for ‘x’, divide both sides by 18:
x < 3.
This means the maximum value of ‘x’ is just under 3. If we take x = 3 for calculation, we get:
- Length of side 1 = 5x = 5 * 3 = 15 cm
- Length of side 2 = 6x = 6 * 3 = 18 cm
- Length of side 3 (longest side) = 7x = 7 * 3 = 21 cm
However, we need to respect the perimeter limitation. The actual maximum value of ‘x’ we could use that stays under the perimeter of 54 cm would be slightly less than 3, which will keep 18x just below 54.
Thus, if we approximate ‘x’ closer to but less than 3, we can say that the longest side would be:
7x where x is slightly less than 3, resulting in a very close value to 21 cm but definitely less than that.
In conclusion, the longest side of the triangle, under the given conditions, is slightly less than 21 cm.